I may have taken one more probability class than you, but the expected value for a woman's taxes over her life is 0. There is no such thing as "the average woman", so whatever point you are trying to make is entirely lost.
And women wouldn't have it any other way than to raise a child alone, but that entirely is besides the point.
Curiously enough, that blog post doesn't bother to post the charts of the immensely high market income gap and consumption gap between men and women in New Zealand. Those wholely explain the tax gap, because well, men are earning and spending much more, 50%-75% more for most cohorts.
As for the income support gap, the authors themselves explain that the gov income support is for poorer women who need the money to supplement their own or their spouses' incomes, especially during child-rearing years.
You seem to be correctly noticing some symptoms, but you're misdiagnosing all the problems at hand.
More people sharing a larger total benefit doesn't imply that the average individual receives a larger benefit.
Raising a child alone is a privilege?